The influence of roughness in the equilibrium problem in lubrication with imposed load

In this article we study a lubricated system consisting on a slider moving over a smooth surface and a known external force (the load) applied upon the slider. The slider moves at constant velocity and close proximity to the surface and the gap is filled by an incompressible fluid (the lubricant). At the equilibrium, the position of the slider presents one degree of freedom to be determined by the balance of forces acting on the system: the load and the total force exerted by the pressure of the lubricant. The pressure distribution is described by a variational inequality of elliptic type known as Swift–Stieber model and based on Reynolds equation. The distance h between the surfaces in a two dimensional domain Ω is given by h η ( x 1 , x 2 , y ) = h 0 ( x 1 , x 2 ) + h 1 ( y ) + η , ( x 1 , x 2 ) ∈ Ω , y ∈ [ 0 , 1 ] where h 0 ( x 1 , x 2 ) ∼ | x 1 | α for α > 0 and h 1 ( y ) ∼ | y − y 0 | β for y being the homogenization variable. The main result of the article quantify the influence of the roughness in the load capacity of the mechanism in the following way: If α < 3 γ for 0 < γ ⩽ 2 α < min { 1 γ − 2 , 3 γ } for γ > 2 then, the mechanism presents finite load capacity, i.e. lim η → 0 ∫ Ω p η < ∞. Infinite load capacity is obtained for γ > 1 and α > 2 / ( γ − 1 ). A one dimensional particular case is given for γ > 3 / 2 with infinite load capacity.